Structure sets vanish for certain bundles over Seifert manifolds
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- by Christopher W. Stark PDF
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Abstract:
Let ${M^{n + 3}}$ be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold ${K^3}$ with an effective circle action of hyperbolic type. Assume that the fiber ${N^n}$ in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set ${S_{\text {TOP}}} ({N^n} \times {D^k},\partial )$ of topological surgery vanishes for sufficiently large $k$. Then the projective class and Whitehead groups of $M$ vanish and ${S_{\text {TOP}}} ({M^{n + 3}} \times {D^k},\partial ) = 0$ if $n + k \geqslant 3$ or if ${K^3}$ is closed and $n = 2$. The $\text {UNil}$ groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that ${\text {UNil}}_j$ vanishes in spite of the failure of Cappell’s sufficient condition.References
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Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 285 (1984), 603-615
- MSC: Primary 57Q10; Secondary 18F25, 57R67
- DOI: https://doi.org/10.1090/S0002-9947-1984-0752493-3
- MathSciNet review: 752493